A native of Martinique like Christophine, Annette is Antoinette's mother, and functions in the novel as an awful shadow of what's to come for her daughter. Unlucky in love – check. Discriminated against for being a Creole woman – check. Manipulated by Christophine (probably) – check. Physically attacks her husband – check. Gets declared insane – check. Gets confined for being insane – check. Annette's death even takes place off stage; we never learn how Annette actually died. We just learn about Annette's funeral in an offhand comment by Antoinette as she drinks some cocoa. The fact that we never learn how Annette's life actually ends is a wink at the novel's ending, which is similarly reticent as to how Antoinette's life ends.
And yes, their names are almost the same – but almost. While Antoinette may have inherited many elements of her mother's unfortunate destiny, the fact that there are still differences between the two women's lives suggests that, despite what all the gossip-mongers say, Antoinette didn't "inherit" her mental illness from her mother, but perhaps other factors – social, historical, cultural, but definitely not genetic or racial – played into their mutual fates.